Proper analysis for the Machines dataset in lme4
On Tue, May 13, 2008 at 6:06 PM, Reinhold Kliegl
<reinhold.kliegl at gmail.com> wrote:
Dear Michael, My following statement was not correct: On Mon, Apr 28, 2008 at 8:38 AM, Reinhold Kliegl <reinhold.kliegl at gmail.com> wrote:
The comparison of m1 and m2 (or m1r and m2r) is conceptually questionable. m1 assumes there are 6 Workers; m2 assumes that there are 18 Workers, that is different groups of 6 persons worked on each of the 3 machines. Presumably, the experimental design decides whether m1 (m1r) or m2 (m2r) is the correct choice.
After reading Douglas Bates's explanation of random effects for the Machines data today, I must add the following correction: mr1: score ~ Machine + (1 | Worker) mr2: score ~ Machine + (1 | Worker/Machine) == score ~ Machine + (1 | Worker) + (1 | Worker:Machine) mr3: score ~ Machine + (Machine | Worker) I wrongly assumed "(1 | Worker/Machine)" would force the six workers to be nested within Machine, that the program would treat them as 3 groups of 6 workers although they are coded 1 to 6. Rather this syntax is shorthand for: "(1 | Worker) + (1 | Worker:Machine)". Viewed this way, mr1 and mr2 can be compared, of course, as can mr1 and mr3; these are nested models; I am not sure about mr2 and mr3.
mr2 and mr3 are nested. It is easiest to see this if mr3 is written in the equivalent form mr3: score ~ Machine + (0 + Machine | Worker) If mr2 were re-expressed in this form the 3 by 3 variance-covariance matrix for the levels of Machine given Worker would have a compound symmetry form in which all the diagonal elements are equal to sigma^2_1 + sigma^2_2 and all the off-diagonal elements are equal to sigma^2_2.